Find the arc length of the graph of the function over the indicated interval.
y = (x⁵/10) - (1/(6x³)) , [1,20]
Thank you so much!!
To find the arc length of the graph of the function over the given interval, we can use the arc length formula for a curve defined by a function:
L = ∫[a,b] √(1 + (f'(x))²) dx
In this case, we need to find the arc length of the graph of the function y = (x⁵/10) - (1/(6x³)) over the interval [1, 20]. To begin, we need to find the derivative of the function y with respect to x, which will help us calculate f'(x):
y = (x⁵/10) - (1/(6x³))
f'(x) = (5x⁴/10) + (1/(2x⁴))
Now, we can substitute the derivative f'(x) back into the arc length formula and integrate over the given interval [1, 20]:
L = ∫[1,20] √(1 + (f'(x))²) dx
L = ∫[1,20] √(1 + ((5x⁴/10) + (1/(2x⁴)))²) dx
To find this integral, we can use numerical methods or software tools such as calculators or computer programs for numerical integration.